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Re: confusing no.2

1.So we can change "put up" to "putting up" "hit" to "hitting"
to make the sentences "Michael put up his posters" "David hit mary" to be noun?

No, we can't. In both of these cases, the -ing verbh form is a verbal. A verbal is a verb form used not as a verb, but as another part of speech. In both of yoyur examples, the -ing verbal is a present participle, used as an adjective. Because it retains verb-like qualities, the participle can take verb modifiers and a direct object. In the first, "putting up his posters" is a participial phrase describing an action of Michael; in the second, "hitting Mary" is a participial phrase describing an action of David.
Another -ing verbal is called a gerund. A gerund is a verbal used as a noun. We'll get to that later.

When you use a real verb, you need a subject and a predicate, so you make a clause.

Michael and David disagree as to whether Michael should put up his posters.

When David hits Mary, it is the worst part of the show.

2.Is this style of writting is suitable to use in formal writting?
If it is not suitable, how can we change the two sentences
above to be more suitable?

3.Can we put " 's " to make it the possessive noun like this?

1 Michael and David disagree about Michael's puttting up his posters

2 David's hitting mary is the worst scene of the show.

The original forms are acceptable in formal writing. A few purists remain who do not accept your original version and insist on the use of the possessive as you indicated in question 3. Historically, the possessive form was the only one deemed acceptable. With the possessive modifier, the -ing verbal becomes a gerund, acting as a noun. However, it became clear that some phrases came out very clumsy with the possessive-gerund combination. So, most now accept the participial analysis that I gave you above.

Both of these second examples are grammatical. I prefer the first one to the second one. The second is a bit clumsy, in my opinion. I prefer it as "David hitting Mary". :wink: